Question
Find the solution of the exponential equation \( e^{2 x+1}=29 \) in terms of logarithms, or correct to four decimal places. \( x=\square \) Question Help: Video
Ask by Chang Coleman. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x \approx 1.1836 \)
Solution
Solve the equation \( e^{2x+1}=29 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(e^{2x+1}=29\)
- step1: Take the logarithm of both sides:
\(\ln{\left(e^{2x+1}\right)}=\ln{\left(29\right)}\)
- step2: Evaluate the logarithm:
\(2x+1=\ln{\left(29\right)}\)
- step3: Move the constant to the right side:
\(2x=\ln{\left(29\right)}-1\)
- step4: Divide both sides:
\(\frac{2x}{2}=\frac{\ln{\left(29\right)}-1}{2}\)
- step5: Divide the numbers:
\(x=\frac{\ln{\left(29\right)}-1}{2}\)
The solution to the exponential equation \(e^{2x+1}=29\) in terms of logarithms is \(x=\frac{\ln(29)-1}{2}\). Correct to four decimal places, the solution is \(x\approx 1.1836\).
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Simplify this solution The Deep Dive
To solve the equation \( e^{2x+1} = 29 \), we start by taking the natural logarithm of both sides: \[ 2x + 1 = \ln(29) \] Next, we isolate \( x \) by rearranging the equation: \[ 2x = \ln(29) - 1 \] Finally, divide by 2: \[ x = \frac{\ln(29) - 1}{2} \] Calculating this gives approximately \( x \approx 0.8951 \) when rounded to four decimal places. This equation reflects how exponential growth can be modeled using logarithms, making it easier to find unknown values. In practical terms, understanding such equations is fundamental in fields ranging from finance to biology, where growth processes are modeled using exponential functions.
